Rings of Generalized and Stable Invariants of Pseudoreflections and Pseudoreflection Groups

Abstract

Let ϱ:G↪GL(n,F) be a representation of a finite groupGover the field F and F[V] the space of polynomial functions onV=Fn. We associate toGan idealJ∞(G)⊂F[V] called the ideal of stable invariants ofG(or more precisely ϱ). If S⊂GL(n,F) is a set of pseudoreflections we associate to S an idealI(S)⊂F[V] called the ideal of generalized invariants of S. WhenGis a pseudoreflection group we investigateI(S) for various choices of S⊂Gand the relation betweenJ∞(G) andI(S). To a representation ϱ:G↪GL(n,F) of a finite group, respectively to a set S⊂GL(n,F) of pseudoreflections, we also associate a ring[formula], respectively[formula]. We show that[formula]is always a polynomial algebra over F, and whenever ϱ(G) is generated by semisimple pseudoreflections S that[formula].